$ D = \left[\begin{array}{rr}1 & -2 \\ -1 & -1\end{array}\right]$ $ B = \left[\begin{array}{rrr}0 & 0 & -2 \\ 3 & 3 & 5\end{array}\right]$ What is $ D B$ ?
Solution: Because $ D$ has dimensions $(2\times2)$ and $ B$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ D B = \left[\begin{array}{rr}{1} & {-2} \\ {-1} & {-1}\end{array}\right] \left[\begin{array}{rrr}{0} & \color{#DF0030}{0} & \color{#9D38BD}{-2} \\ {3} & \color{#DF0030}{3} & \color{#9D38BD}{5}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ D$ , with the corresponding elements in column $j$ of the second matrix, $ B$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ D$ with the first element in ${\text{column }1}$ of $ B$ , then multiply the second element in ${\text{row }1}$ of $ D$ with the second element in ${\text{column }1}$ of $ B$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{1}\cdot{0}+{-2}\cdot{3} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ D$ with the corresponding elements in ${\text{column }1}$ of $ B$ and add the products together. $ \left[\begin{array}{rrr}{1}\cdot{0}+{-2}\cdot{3} & ? & ? \\ {-1}\cdot{0}+{-1}\cdot{3} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ D$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ B$ and add the products together. $ \left[\begin{array}{rrr}{1}\cdot{0}+{-2}\cdot{3} & {1}\cdot\color{#DF0030}{0}+{-2}\cdot\color{#DF0030}{3} & ? \\ {-1}\cdot{0}+{-1}\cdot{3} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{1}\cdot{0}+{-2}\cdot{3} & {1}\cdot\color{#DF0030}{0}+{-2}\cdot\color{#DF0030}{3} & {1}\cdot\color{#9D38BD}{-2}+{-2}\cdot\color{#9D38BD}{5} \\ {-1}\cdot{0}+{-1}\cdot{3} & {-1}\cdot\color{#DF0030}{0}+{-1}\cdot\color{#DF0030}{3} & {-1}\cdot\color{#9D38BD}{-2}+{-1}\cdot\color{#9D38BD}{5}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}-6 & -6 & -12 \\ -3 & -3 & -3\end{array}\right] $